Problem 46 - College Algebra with Corequisite Support

Question

For the following exercises, write the first four terms of the sequence. $$ a_{n}=\frac{100 \cdot n}{n(n-1) !} $$

Step-by-Step Solution

Answer

The first four terms are: 100, 100, 50, and \( \frac{50}{3} \).

Step 1: Calculate the first term of the sequence

To find the first term, substitute \( n = 1 \) into the formula: \[ a_1 = \frac{100 \cdot 1}{1 \cdot (1-1)!}\] Simplifying, we have:\[ a_1 = \frac{100}{1 \cdot 1} = 100 \] Therefore, \( a_1 = 100 \).

Step 2: Calculate the second term of the sequence

Substitute \( n = 2 \) into the formula: \[ a_2 = \frac{100 \cdot 2}{2(2-1)!} \] Simplifying, we have:\[ a_2 = \frac{200}{2 \cdot 1} = 100 \] Thus, \( a_2 = 100 \).

Step 3: Calculate the third term of the sequence

Substitute \( n = 3 \) into the formula: \[ a_3 = \frac{100 \cdot 3}{3(3-1)!} \] Simplifying, we have:\[ a_3 = \frac{300}{3 \cdot 2} = 50 \] Thus, \( a_3 = 50 \).

Step 4: Calculate the fourth term of the sequence

Substitute \( n = 4 \) into the formula: \[ a_4 = \frac{100 \cdot 4}{4(4-1)!} \] Simplifying, we have:\[ a_4 = \frac{400}{4 \cdot 6} = \frac{400}{24} \] Therefore, \( a_4 = \frac{50}{3} \).

Key Concepts

Factorial

Factorials are an essential concept when dealing with sequences and calculations involving permutations and combinations. In mathematics, the factorial of a non-negative integer, denoted as \( n! \), is the product of all positive integers less than or equal to \( n \). For example:

• \( 3! = 3 \times 2 \times 1 = 6 \)
• \( 4! = 4 \times 3 \times 2 \times 1 = 24 \)

Factorials grow very quickly in value as the number increases. One important case to remember is that \( 0! = 1 \) by definition. This helps simplify expressions with factorials where \( n \) could take the value of 0, as seen in calculating the first term of the sequence in the original problem.Understanding factorials is crucial since they often appear in formulas related to sequences, where patterns or arrangements are involved.

Sequence Terms

A sequence is an ordered set of numbers, and each number is regarded as a term. Learning how to find sequence terms is critical in understanding mathematical sequences. Each term can often be determined by a specific formula, such as the one given in the problem:\[ a_n = \frac{100 \cdot n}{n(n-1)!} \]To find the terms of the sequence, substitute successive integer values for \( n \):

• First term \( a_1 \): Replace \( n \) with 1.
• Second term \( a_2 \): Replace \( n \) with 2.
• And so on...

Each substitution yields a numerical value, helping to build the sequence one term at a time. These terms provide patterns or insights into the nature of the sequence, useful in various mathematical and real-world applications.

Simplification

Simplification involves reducing expressions to their simplest form, making them easier to understand and work with. In arithmetic sequences, simplification is crucial to distill equations or expressions to their basic components. Consider the original sequence expression:\[ a_n = \frac{100 \cdot n}{n(n-1)!} \]Each calculation step involves simplifying both the factorial and the expression:- Calculate factorial \((n-1)!\), which involves multiplying all integers up to \((n-1)\).- Conduct operations such as division or multiplication to reduce the resulting fraction to its simplest form.This simplification process converts complex expressions into easily interpretable results, making the sequence terms clearer and more manageable for further analysis or application.

Mathematical Formulae

Mathematical formulae are expressions that show relationships between different quantities, and they are key to solving various mathematical problems. In the exercise, formulae are used to define a sequence:\[ a_n = \frac{100 \cdot n}{n(n-1)!} \]Formulae help in:

• Defining sequences: They provide a general rule which, when applied, yields the terms of a sequence.
• Simplifying complex problems: Break down problems into simple calculations using the formula.
• Accurately calculating specific values: By substituting particular values of \( n \), the sequence's terms are calculated.

Understanding formulae involves knowing how to apply operations and basic algebraic rules. These formulaic representations are powerful tools in both theoretical and applied mathematics, aiding in exploration and problem-solving across diverse fields.